1. Balanced Chemical Equation:

The balanced chemical equation for the reaction is:

3 NO₂ (g) + H₂O (l) → 2 HNO₃ (aq) + NO (g)

2. Mole-to-Mole Ratio:

From the balanced equation, we see that 3 moles of NO₂ react to produce 1 mole of NO.

3. Convert Molecules to Moles:

* We have 5.00 x 10²² molecules of NO.

* Avogadro's number tells us there are 6.022 x 10²³ molecules in one mole.

* Moles of NO = (5.00 x 10²² molecules) / (6.022 x 10²³ molecules/mol) = 0.0830 mol NO

4. Calculate Moles of NO₂:

* Using the mole ratio from the balanced equation:

Moles of NO₂ = 0.0830 mol NO * (3 mol NO₂ / 1 mol NO) = 0.249 mol NO₂

5. Convert Moles of NO₂ to Grams:

* The molar mass of NO₂ is 14.01 g/mol (N) + 2 * 16.00 g/mol (O) = 46.01 g/mol

* Grams of NO₂ = 0.249 mol NO₂ * 46.01 g/mol = 11.4 g NO₂

Therefore, you need approximately 11.4 grams of nitrogen dioxide to react with water to produce 5.00 x 10²² molecules of nitrogen monoxide.